chapter 2_dsp1_co thuc_full
TRANSCRIPT
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CHAPTER 2:
DISCRETE-TIME SIGNALS & SYSTEMS
Lesson #4: DT signals
Lesson #5: DT systems
Lesson #6: DT convolution
Lesson #7: Difference equation models
Lesson #8: Block diagram for DT LTI systems
Duration: 9 hrs
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Lecture #4
DT signals
1. Representations of DT signals
2. Some elementary DT signals
3. Simple manipulations of DT signals
4. Characteristics of DT signals
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Converting a CT signal into a DT signal by sampling: given xa(t) to
be a CT signal, xa(nT) is the value ofxa(t) at t = nT DT signal is
defined only forn an integer
n),n(x)nT(x)t(x anTta
-2T -T 0 T 2T 3T 4T 5T 6T 7T . . . nT
t
Sampled signals
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n -1 0 1 2 3 4
x[n] 0 0 1 4 1 0
1.Functional representation
n,0
2n,4
3,1n,1
]n[x
Representations of DT signals
2.Tabular representation
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3.Sequence representation
Representations of DT signals
1,4,1,0][nx
-1 0 1 2 3 4 5 n
4.Graphical representation 4
1 1
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Lecture #4
DT signals
1. Representations of DT signals
2. Some elementary DT signals
3. Simple manipulations of DT signals
4. Classification of DT signals
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1. Unit step sequence
2. Unit impulse signal
3. Sinusoidal signal
4. Exponential signal
Some elementary DT signals
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1 0[ ]
0 0
nu n
n
Unit step
-1 0 1 2 3 4 5 6 n
1 1 1
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Time-shifted unit step
0
0
0nn,0
nn,1]nn[u
0 -n0-1 n0 n0+1 n
For n0 > 0
1 1 1
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Time-shifted unit step
0
0
0nn,0
nn,1]nn[u
-n0-1 n0 n0+1 0 n
For n0 < 0
1 1 1
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Unit impulse
1 0[ ]
0 0
nn
n
-2 -1 0 1 2 n
1
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Time-shifted unit impulse
0
0
0nn,0
nn,1]nn[
For n0 > 0
0 n0-1 n0 n0+1 n
1
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Time-shifted unit impulse
0
0
0nn,0
nn,1]nn[
n0-1 n0 n0+1 0 n
1
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Relation between unit step andunit impulse
]n[x]nn[]n[x
]nn[]n[x]nn[]n[x
]1n[u]n[u]n[
]k[]n[u
0
n
0
000
n
k
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Sinusoidal signal
n),nF2cos(A
n),ncos(A)n(x
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Exponential signal
nCa]n[x
1. IfCand aare real, then x[n]is a real exponential
a > 1 growing exponential
0 < a < 1 shrinking exponential
-1 < a < 0 alternate and decay
a < -1 alternate and grows
2. IfCor aor both is complex, then x[n]is a complexexponential
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An example of real exponential signal
nnx )2.1)(2.0(][
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An example of complexexponential signal
nj
enx 6121
2][
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Periodic exponential signal
Recall:A DT sinusoidal signal is periodic only if its
frequency is a rational number
Consider complex exponential signal:
It is also periodic only if its frequency is a rational number:
N
kor
N
kF
2
00
)sin()cos(][ 000 njnCCenx
nj
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Fundamental period
The fundamental period can be found as
Where k is the smallest integer such that N is an integer
Step 1: Is rational?
Step 2: If yes, then periodic; reduce to
0
2kN
spo
cycles
N
k
int#
#
2
0
2
0
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Examples
Determine which of the signals below are periodic. For the
ones that are, find the fundamental period and
fundamental frequency nj
enx6
1 ][
612
22
12
12
1
)2(62
0
0
N
N
N
k One cycle in 12 points
: fundamental period
: fundamental frequency
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Examples
Determine which of the signals below are periodic. For theones that are, find the fundamental period and
fundamental frequency
15
3
sin][2 nnx
510
22
10
10
3
)2(5
3
2
0
N
N
N
k 3 cycles in 10 points
: fundamental period
: fundamental frequency 0because k 1
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Examples
Determine which of the signals below are periodic. For the
ones that are, find the fundamental period and
fundamental frequency
)2cos(][3 nnx
0 = 2 0/2 = 1/ is irrational x3[n] is not periodic
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Examples
Determine which of the signals below are periodic. For the
ones that are, find the fundamental period and
fundamental frequency
)2.1cos(][4 nnx
5
22
5
5
3
2
2.1
2
0
N
N
N
k 3 cycles in 5 points
: fundamental period
: fundamental frequency 0because k 1
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Lecture #4
DT signals
1. Representations of DT signals
2. Some elementary DT signals
3. Simple manipulations of DT signals
4. Classification of DT signals
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Adding and subtracting signals
Transformation of time:
- Time shifting
- Time scaling
- Time reversal
Transformation of amplitude:
- Amplitude shifting
- Amplitude scaling
- Amplitude reversal
Simple manipulations of DT signals
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Do it point by point
Can do using a table, or graphically, or by
computer program
Example: x[n] = u[n] u[n-4]
Adding and Subtracting signals
n =4
x[n] 0 1 1 1 1 0
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x[n] x[n - k]; k is an integer
k > 0: right-shift x[n] by |k| samples
(delay of signal)
k < 0: left-shift x[n] by |k| samples
(advance of signal)
Time shifting a DT signal
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Examples of time shifting
-1 0 1 2 3 4 n
4
1 1
x[n]
-1 0 1 2 3 4 n
4
1 1
x[n-2]
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Examples of time shifting
-1 0 1 2 3 4 n
4
1 1
x[n]
-1 0 1 2 3 4 n
4
1 1
x[n+1]
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Time scaling a DT signal
x[n] y[n] = x[an]
|a| > 1: speed up by a factor of a
a must be an integer
|a| < 1: slow down by a factor of a
a = 1/K; K must be an integer
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Examples of time scaling
-2 -1 0 1 2 n
4
-1 0 1 2 n
1
x[2n]x[2n+1]
-1 0 1 2 3 4 n
4
1 1
x[n]
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Given x[n]
Examples of time scaling
w1[n] = x[2n]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
1
2
2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
What does w1[n/2]look like? Just look like x[n]!
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x[n] x[-n]
Flip a signal about the vertical axis
Time reversal a DT signal
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Examples of time reversal
-1 0 1 2 3 4 n
4
1 1
x[n]
-3 -2 -1 0 1 2 3 4 n
4
1
x[-n]
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x[n] y[n] = x[-n-k]
Method 1: Flip first, then shift
Method 2: Shift first, then flip
Combining time reversal and time shifting
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4
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Example
Method 2-1 0 1 2 3 4 5 n
1
4
x[n]
-4 -3 -2 -1 0 1 2 3 4 5 n
w[-m] = x[-(n+2)] = x[-n-2]
x[n+2] = w[m]
-2 -1 0 1 2 3 4 5 6 n
-2 -1 0 1 2 3 4 5 6 7 8 m
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x[n] y[n] = x[an-b]
Method 1: time scale then shift
Method 2: shift then time scale
Be careful!!! For some cases, method 1 or 2 doesnt work.To make sure, plug values into the table to check
Combining time shifting and time scaling
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4
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Example
Method 1
x[-2n] = w[n]
-4 -3 -2 -1 0 1 2 3 4 5 n
-1 0 1 2 3 4 5 n
1
4
x[n]
-4 -3 -2 -1 0 1 2 3 4 5 n
w[n-1] = x[-2(n-1)] = x[-2n+2]
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x[n] y[n] = x[an-b]
Method 2: shift then time scale
Ex. Find y[n] = x[2-2n]
y[n] = x[-2(n-1)]
x[n]
x[n-1] = w[m]
w[-2m] = x[-2(n-1)]
Example Method 2
Delayby 1
Timescaleby -2
4
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Example
Method 2 -1 0 1 2 3 4 5 n
1
x[n]
-4 -3 -2 -1 0 1 2 3 4 5 n
w[-2m] = x[-2(n-1)] = x[-2n+2]
x[n-1] = w[m]
-1 0 1 2 3 4 5 n
-2 -1 0 1 2 3 4 m
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Example
y[n] = x[2n-3]??
n x[n] y[n]-1 0 0
0 0 0
1 1 02 4 1
3 1 1
4 0 0
-1 0 1 2 3 4 n
4
1 1
x[n]
-1 0 1 2 3 4 n
1y[n]
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Find
Exercise
[ ] ( [ 1] [ 5])( [2 ])x n u n u n nu n
-1
21
-1
x[n]
0 1 2 3 4 5 n
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Lecture #4
DT signals
1. Representations of DT signals
2. Some elementary DT signals
3. Simple manipulations of DT signals
4. Characteristics of DT signals
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Symmetric (even) and anti-symmetric (odd)signals
Energy and power signals
Characteristics of DT signals
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A DT signal xe[n] is evenif
And the signal xo[n] is oddif
Any DT signal can be expressed as the sum of an even
signal and an odd signal:
Even and odd signals
Even [ ] [ ]e ex n x n
Odd [ ] [ ]o o
x n x n
12
[ ] ( [ ] [ ])ex n x n x n
12
[ ] ( [ ] [ ])ox n x n x n
[ ] [ ] [ ]e ox n x n x n
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How to find xe[n]and xo[n]from a given x[n]?
Step 1: find x[-n]
Step 2: find
Step 3: find
Even and odd signals
12
[ ] ( [ ] [ ])e
x n x n x n
12
[ ] ( [ ] [ ])o
x n x n x n
Example
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Given x[n]
Example
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8
1
2
Find x[-n]
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1
2
Example
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Find x[n] + x[-n]
Example
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8
1
2
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1
3/2
1/2
Find xe[n]
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Determine which of the signals below are energy signals?
Which are power signals?
Examples
(a) Unit step1 0[ ]0 0
nu nn
0
22 1][nn
nxE
02/112
1lim1
12
1lim][
12
1lim
0
22
N
N
Nnx
NP
N
N
nN
N
NnN
Unit step is a power signal
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E l
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Determine which of the signals below are energy signals?
Which are power signals?
Examples
(c) ])4n[u]n[u(n4
cos]n[x
]3[2
2]1[
2
2][
0
304
cos][ nnn
otherwise
nnnx
242
421
22
221][
22
2
n
nxE
x[n] is an energy signal
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I t ti f DT t
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y[n] = y1[n] + y
2[n] = T
1{x[n]} + T
2{x[n]} = (T
1+ T
2){x[n]}
= T{x[n]}
y[n] = T{x[n]}: notation for the total system
Interconnection of DT systems
System1T1{ }
x[n] y[n]
System2T2{ }Parallel
connection
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y[n] = T2{y1[n]} = T2{T1{x[n]}} = T{x[n]}
y[n] = T{x[n]}: notation for the total system
Interconnection of DT systems
System1T1{ }
x[n] y[n]System2T2{ }
Cascade connection
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Lecture #5
DT systems
1. DT system
2. DT system properties
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Memory
Invertibility
Causality
Stability
Linearity
Time-invariance
DT system properties
M
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y[n0] = f(x[n0]) system is memoryless (static)
Otherwise, system has memory (dynamic), meaning that its
output depends on inputs rather than just at the time of the
output
Ex:
a) y[n] = x[n] + 5: is memoryless
b) y[n]=(n+5)x[n]: is memoryless
c) y[n]=x[n+5]: has memory
Memory
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Invertibility
A system is said to be invertible if distinct inputs
result in distinct outputs
Ex.: y[n] = |x[n]| is not invertible
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Invertibility
Ti[T(x[n])] = x[n]
T() Ti()
x[n] x[n]
System Inverse system
A system is said to be invertible if distinct inputs
result in distinct outputs
Examples for invertibility
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Examples for invertibility
Determine which of the systems below are invertible
a) Unit advance y[n] = x[n+1]
b) Accumulator
c) Rectifier y[n] = |x[n]|
n
k
kxny ][][
1
][]1[
n
kkxny y[n] y[n-1] = x[n] is inverse system
is invertible
is invertible
is not invertible
y[n-1] = x[n] is inverse system
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The output of a causal system (at each time) does not
depend on future inputs
All memoryless systems are causal
All causal systems can have memory or not
Causality
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Examples for causality
Determine which of the systems below are causal:
a) y[n] = x[-n]
b) y[n] = (n+1)x[n-1]
c) y[n] = x[(n-1)2]
d) y[n] = cos(w0n+x[n])
e) y[n] = 0.5y[n-1] + x[n-1]
non causal
non causal
causal
causal
causal
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Examples for stability
Determine which of the systems below are BIBO stable:
a) A unit delay system
b) An accumulator
c) y[n] = cos(x[n])
d) y[n] = ln(x[n])
e) y[n] =exp(x[n])
stable
stable
stable
unstable
unstable
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Scaling signals and adding them, then processing through the system
same asProcessing signals through system, then scaling and adding them
Linearity
If T(x1[n]) = y1[n] and T(x2[n]) = y2[n]
T(ax1[n] + bx2[n]) = ay1[n] + by2[n]
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If you time shift the input, get the same output, but with the
same time shift
The behavior of the system doesnt change with time
Time-invariance
If T(x[n]) = y[n]
then T(x[n-n0]) = y[n-n0]
E l f li it d ti
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Examples for linearity and time-invariance
Determine which of the systems below are linear, whichones are time-invariant
a) [ ] [ ]y n nx n
Linear
Not time-invariant
E l f li it d ti
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Examples for linearity and time-invariance
Determine which of the systems below are linear, wichones are time-invariant
b) ]n[x]n[y2
Non-linear
Time-invariant
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E l f DT t ti
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Example for DT system properties
Given the system below:
a) It is memoryless
][5.1
5.2
][
2
nxn
n
ny
b) It is invertible its inverse system is: ][5.2
5.1][
2
nyn
nnx
c) It is causal
E l f DT t ti
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0 100 200 300 400 500 600 700 800 900 10000
1
2
3
4
5
6
7
8
9
Example for DT system properties
Given the system below: ][5.1
5.2
][
2
nxn
n
ny
c) It is stable
For |x[n]| M then |y[n]| 9M
2
5.1
5.2
n
n
E l f DT t ti
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Example for DT system properties
Given the system below: ][5.1
5.2
][
2
nxn
n
ny
d) For n = 0 y[0] = 2.778x[0]
For n = -1 y[-1] = 9x[-1] time invariant
e) Superposition linear
][][
][5.1
5.2][
5.1
5.2
][][5.1
5.2][][
2211
2
2
21
2
1
2211
2
2211
nyanya
nxn
nanx
n
na
nxanxan
nnxanxa
Lecture #6
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Lecture #6
DT convolution
1. DT convolution formula
2. DT convolution properties
3. Computing the convolution sum
4. DT LTI properties from impulse response
Computing the response of DT LTI
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Method 1:based on the direct solution of the input-output equation
for the system
Method 2:
Decompose the input signal into a sum of elementary signals
Find the response of system
to each elementary signal
Add those responses to obtain
the total response of the system
to the given input signal
Computing the response of DT LTIsystems to arbitrary inputs
k
kk
kk
k
kk
nycnynx
nynx
nxcnx
][][][
][][
][][
DT convolution formula
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Convolution: an operation between the input signal to a
system and its impulse response, resulting in the output signal
CT systems: convolution of 2 signals involves integrating the
product of the 2 signals where one of signals is flipped and
shifted
DT systems: convolution of 2 signals involves summing theproduct of the 2 signals where one of signals is flipped and
shifted
DT convolution formula
I l t ti f DT i l
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We can describe any DT signal x[n] as:
Example:
Impulse representation of DT signals
[ ] [ ] [ ]k
x n x k n k
-1 0 1 2 3 n
x[n]
-1 0 1 2 n
x[0][n-0]
-1 0 1 2 n
x[1][n-1]
-1 0 1 2 n
x[2][n-2]+ +
Impulse response of DT systems
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Impulse response: the output results, in response to a unit impulse
Denotation: hk[n]: impulse response of a system, to an impulse at
time k
Impulse response of DT systems
Time-invariantDT system
Time-invariant
DT system
[n]
[n-k]
h[n]
h[n-k]
Remember:the impulse response is a sequence of values that may
go on forever!!!
Response of LTI DT systems to
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p yarbitrary inputs
LTI DT system[n-k] h[n-k]
LTI DT system
[ ] [ ] [ ]k
x n x k n kk
knhkxny ][][][
Notation:y[n] = x[n] * h[n]
Convolution sum
Convolution sum in more details
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Convolution sum in more details
y[0] = x[0] * h[0]
= + x[-2]h[2] + x[-1]h[1] + x[0]h[0]
+ x[1]h[-1] + x[2]h[-2] + +
The general output:
y[n] = + x[-2]h[n+2] + x[-1]h[n+1] + x[0]h[n]
+ x[1]h[n-1] + x[2]h[n-2] + + x[n-1]h[1]
+ x[n]h[0] + x[n+1]h[-1] + x[n+2]h[-2]
Note:the sum of the arguments in each term is always n
Lecture #6
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Lecture #6
DT convolution
1. DT convolution formula
2. DT convolution properties
3. Computing the convolution sum
4. DT LTI properties from impulse response
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[n] * x[n] = x[n]
[n-m] * x[n] = x[n-m]
[n] * x[n-m] = x[n-m]
Commutative law
Associative law
Distributive law
Convolution sum properties
Commutative law
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Commutative law
][*][][*][ nxnhnhnx
h[n]x[n] y[n]
x[n]h[n] y[n]
Associative law
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Associative law
])[*][(*][][*])[*][( 2121 nhnhnxnhnhnx
h1[n]x[n] y[n]
h2[n]
h2[n]x[n] y[n]
h1[n]
h1[n]*h2[n]
x[n] y[n]
Distributive law
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Distributive law
])[*][(])[*][(])[][(*][ 2121 nhnxnhnxnhnhnx
h1[n] + h2[n]x[n] y[n]
h1[n]
x[n] y[n]
h2[n]
Lecture #6
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Lecture #6
DT convolution
1. DT convolution formula
2. DT convolution properties
3. Computing the convolution sum
4. DT LTI properties from impulse response
Computing the convolution sum
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1. Foldh[k]about k = 0, to obtain h[-k]
2. Shifth[-k]by n0to the right (left) ifn0is positive (negative), toobtain h[n0-k]
3. Multiplyx[k]and h[n0-k]for all k, to obtain the product
x[k].h[n0-k]
4. Sum up the product for all k, to obtain y[n0]Repeat from 2-4 fof all of n
kkknhkxnyknhkxny ][][][][][][ 00
The length of the convolution sum result
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The length of the convolution sum result
Suppose:
Length of x[k] is Nx N1 k N1 + Nx 1
Length of h[n-k] is Nh N2 n-k N2 + Nh 1
N1 + N2 n N1 + N2 + Nx + Nh 2
Length of y[n]:
Ny = Nx + Nh 1
[ ] [ ] [ ] [ ] [ ]k
y n x n h n x k h n k
Example 1
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Example 1
Find y[n] = x[n]*h[n] where
[ ] [ 1] [ 3] [ ]x n u n u n n [ ] 2 [ ] [ 3]h n u n u n
n
n
x[n]
h[n]
-1 0 1 2 3
-1 0 1 2 3
Ex1 (cont)
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h[-k]
h[k]
x[k]
-1 0 1 2 3 k
-1 0 1 2 3 k
-2 -1 0 1 ky[0] = 6;
Ex1 (cont)
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h[-k]
x[k]
-1 0 1 2 3 k
-2 -1 0 1 k
-4 -3 -2 -1 0 k
h[-1-k]
y[-1] = 2;
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Ex1 (cont)
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( )
y[2] = 8;
h[-k]
x[k]
-1 0 1 2 3 k
-2 -1 0 1 k
h[2-k]
-2 -1 0 1 2 k
Ex1 (cont)
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( )
y[3] = 4;
h[-k]
x[k]
-1 0 1 2 3 k
-2 -1 0 1 k
h[3-k]
-2 -1 0 1 2 k
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Example 2
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Find y[n] = x[n]*h[n] where [ ] [ ]n
x n a u n [ ] [ ]h n u n
Try it both ways (first flip x[n] and do the convolution and then flip
h[n] and do the convolution). Which method do you prefer?
1||
1||1
0
0 aif
aifa
a
a
n
nn
n
a
aaa
nnn
n
nn
n
1
1)1( 01
0
1
0
Remember!
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Example 3
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Find y[n] = x[n]*h[n] where x[n] = bnu[n] and h[n] = anu[n+2]
|a| < 1, |b| < 1, a b
b
a
ba
b
ab
b
ababnyn
nyn
n
nn
k
k
nkn
k
kn
1
1
][2
0][:2
3
2
22
]2[1
][1
1122
nuab
babany
nn
Therefore
Factor out bn toget form you know
Example 4
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Compute output of a system with impulse response
h[n] = an
u[n-2], |a| < 1when the input is x[n] = u[-n]
a
aanyn
a
aann
n
nk
k
k
k
1][:2
1
][:22
2
]3[1
]2[1
][2
nua
anu
a
any
nTherefore
Flipping x[n] because it is simpler
A constant Varies with n
Lecture #6
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DT convolution
1. DT convolution formula
2. DT convolution properties
3. Computing the convolution sum
4. DT LTI properties from impulse response
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Calculation of the impulse response
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Applying the unit impulse function to the input-output equation
Ex.: ]1n[x2
1]n[x21]1n[y
41]n[y
]1[)8/5()4/1(][)2/1(][ 1 nunny n
Suppose this
system is causal
2
1
0
4
1
8
5
8
5
4
1
4
1]2[
2
1]3[
2
1]2[
4
1]3[
4
1
8
5]1[
2
1]2[
2
1]1[
4
1]2[
41
85
85
21
21
41]0[
21]1[
21]0[
41]1[
2
1]1[
2
1]0[
2
1]1[
4
1]0[
xxhh
hh
xhh
hh
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Examples
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a p es
1. Is h[n] = 0.5nu[n] BIBO stable? Causal?
2. Is h[n] = 3nu[n] BIBO stable? Causal?
3. Is h[n] = 3nu[-n] BIBO stable? Causal?
Stable
Not stable
Stable
Causal
Causal
Not causal
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Linear constant coefficient difference equations
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General form:
q
][...]1[][][...]1[][ 101 MnxbnxbnxbNnyanyany MN
1a,]rn[xb]kn[ya 0
M
0r
r
N
0k
k
N, M: non-negative integers
N: order of equation
ak, br: real constant coefficients
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Components for block diagram
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EX: an averaging system y[n] = 0.5(x[n] + x[n-1])
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Direct form II realization of a system
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-a1
Z-1
-a2
Z-1
-aN
b0
b1
b2
bN
x[n] y[n]
Suppose
M = N
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HW
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Prob.1
a) 9.1a (iii)
b) 9.1a (iv)
c) 9.2a (i)
d) 9.2a (ii)
e) 9.2a (vi)
f) 9.5a
Prob.2
a) 9.13a (iii)
b) 9.15d
c) 9.17a
d) 9.17b
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